https://orcid.org/0000-0003-3430-2464
Correction
12 Dec 2024: Aljohani HM, Bandar SA, Al-Mofleh H, Ahmad Z, El-Morshedy M, et al. (2024) Correction: A new asymmetric extended family: Properties and estimation methods with actuarial applications. PLOS ONE 19(12): e0315953. https://doi.org/10.1371/journal.pone.0315953 View correction
Figures
Abstract
In the present work, a class of distributions, called new extended family of heavy-tailed distributions is introduced. The special sub-models of the introduced family provide unimodal, bimodal, symmetric, and asymmetric density shapes. A special sub-model of the new family, called the new extended heavy-tailed Weibull (NEHTW) distribution, is studied in more detail. The NEHTW parameters have been estimated via eight classical estimation procedures. The performance of these methods have been explored using detailed simulation results which have been ordered, using partial and overall ranks, to determine the best estimation method. Two important risk measures are derived for the NEHTW distribution. To prove the usefulness of the two actuarial measures in financial sciences, a simulation study is conducted. Finally, the flexibility and importance of the NEHTW model are illustrated empirically using two real-life insurance data sets. Based on our study, we observe that the NEHTW distribution may be a good candidate for modeling financial and actuarial sciences data.
Citation: Aljohani HM, Bandar SA, Al-Mofleh H, Ahmad Z, El-Morshedy M, Afify AZ (2022) A new asymmetric extended family: Properties and estimation methods with actuarial applications. PLoS ONE 17(10): e0275001. https://doi.org/10.1371/journal.pone.0275001
Editor: Hubert Amu, University of Health and Allied Sciences, GHANA
Received: July 12, 2022; Accepted: September 8, 2022; Published: October 6, 2022
Copyright: © 2022 Aljohani et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The details for accessing the relevant data are available in the article.
Funding: The Taif University Researchers Supporting Project provided support for this study in the form of a grant awarded to HMA (TURSP-2020/279). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Probability distributions have been extensively used to model real-life data is several applied areas. For example, glass fiber, breaking stress of carbon fiber, and gauge lengths data [1] and wind speed data [2]. Speaking broadly, the heavy-tailed models play a vital role in analyzing and modelling data in several applied areas such as banking, risk management, financial and actuarial sciences, and economic, among others. The quality of statistical procedures mainly depends on the considered probability model of the studied phenomenon.
Usually the insurance and financial data sets are positive or right skewed [3, 4], or unimodal shaped [5], and having heavy tails [6]. [7] showed that right-skewed data sets can be modeled using skewed distributions. Therefore, some unimodal and right-skewed distributions were employed to analyze such data sets (see, e.g. [8–10]). The recent developments have been suggested using several approaches including (i) composition of two or more distributions, (ii) transformation of variables and (iii) compounding of distributions. More details can be explored in [11].
Some notable recent proposed families in the statistical literature include the odd Chen-G [12], alpha-power Marshall–Olkin-G [13], type-I half-logistic Weibull-G [14], new flexible generalized class [15], additive odd-G [16], Marshall–Olkin–Weibull-H families [17].
In this paper a flexible class of distributions, called the new extended heavy-tailed-F (NEHT-F) family is studied. The new extended heavy-tailed Weibull (NEHTW) distribution is studied in detail as a special case of this family. The proposed NEHTW model has greater flexibility and it can provide adequate fits in modelling heavy-tailed real data encountered in insurance field and other applied fields. The NEHTW parameters are estimated using eight different estimation methods. The simulation results are conducted to evaluate the performance of these methods of estimation and to choose the best estimation method for the NEHTW parameters based on ranks.
The proposed NEHT-F family are motivated by the following desirable properties. (i) Its special models can provide right-skewed, unimodal, and reversed-J densities which are suitable for modelling heavy-tailed real data encountered in insurance science and many other applied sciences. Furthermore, the hazard rate function (hrf) of the special models of the NEHT-F family can be constant, increasing, modified bathtub, decreasing, J-shaped, and reversed-J shaped hazard rates; (ii) The density functions of its special models can be bimodal, unimodal, symmetric and asymmetric shapes; (iii) The cdf, pdf, hrf of the proposed family have simple closed form expressions. Hence the special sub-models of the NEHT-F family can be used in analyzing and modelling censored data; (iv) The NEHTW distribution, as a special sub-model of NEHT-F family, includes the Weibull distribution as a special case and it can provide adequate fit for heavy-tailed insurance data; (v) We discussed and derived two important risk measure, VaR and TVaR, for the NEHTW model and validate them based on simulations and real data applications, showing that the NEHTW has a heavier tail than the Weibull distribution.
Furthermore, the NEHTW parameters are estimated via eight classical estimation procedures. The performance of these approaches are explored using detailed simulation results which have been ordered, using partial and overall ranks, to determine the best estimation method. This will develop a guideline for choosing the best estimation method for the NEHTW distribution, which we think would be of interest to engineers.
The rest of this work was presented as follows. The NEHT-F family is introduced in Section 2. Some special sub-models of the NEHT-F family are provided in Section 3. The expressions for VaR and TVaR of the NEHTW model along with a simulation study are discussed in Section 4. In Section 5, some mathematical properties of the NEHT-F family are reported. The maximum likelihood estimators of the new family parameters are obtained in Section 6. Eight different classical estimators are discussed to estimate the NEHTW parameters in Section 7. In Section 8, the performance of the introduced estimators is evaluated via Monte Carlo simulations. Two heavy-tailed real-life data sets from the insurance sciences are analysed in Section 9. Some conclusions are provided in Section 10.
2 Genesis of the proposed family
The heavy-tailed distributions have heavier right-tails than those of the exponential one. That is, for a baseline G(x), cumulative distribution function (cdf), the following formula holds
(1)
The interested readers can refer to [18–20].
In the present work, an alternative approach to (1), we suggest a empirical approach to find heavy-tailed distributions. The proposed approach is based on the Monte Carlo simulation of the actuarial measures, namely value at risk (VaR) and tail value at risk (TVaR).
Let X represent the profit and loss distribution (profit positive and loss negative). The VaR at level q ∈ (0, 1) is the lowest number, where the probability that Y = −X does not exceed y is at least 1 − q. Mathematically, VaRq(X) is the (1 − q)th quantile of Y, i.e.,
(2)
for detail see [21, 22].
The TVaR is also referred to as tail conditional expectation (TCE). The TVaR determines the expected value of the losses under the condition that an event outside a certain probability level was occurred.
The TVaR is specified by
(3)
where xq is the upper quantile given by
for some details see e.g., [21, 23–25].
To apply the simulation approach based on (2) and (3), we first introduce a new family of distributions and then provide a simulation study of the VaR and TVaR for a special sub-model of the proposed family to show empirically the heaviness of the tail of the proposed distribution.
Recently, [26] proposed a new wider family called extended Cordeiro and de Castro (ECC) family which is defined by the cdf
(4)
where
and p ∈ (0, 1). In the ECC family, the space of the parameter p is restricted to (0,1), hence the family with cdf (4) can not have enough flexibility to counter complex forms of data. Furthermore, the estimation of parameters of the special models of the ECC family as well as the computation of several distributional characteristics become very difficult due to the three additional parameters of the ECC family. Therefore, in this paper a flexible class of distributions, called the NEHT-F family is studied. The new family is proposed by keeping constant θ = 1 in (4) (to reduce the number of parameters), and instead of parameter p, we use a new parameter σ > 0 with unbounded upper bound unlike the upper limit of p. The new extended class of heavy-tailed distributions is introduced via the T-X family approach of [27].
The T-X family approach is specified by the cdf
(5)
where V[F(x;ξ)] satisfies three conditions as illustrated in [27].
The pdf corresponding to (5) is
Let w(t) be the exponential density, i.e., w(t) = ηe−ηt. Then, we replace the upper limit of (5) by , we obtain the cdf of the proposed family. If a random variable (rv) X follows the proposed NEHT-F family, then its cdf takes the form
(6)
where
and F(x; ξ) is the baseline cdf with a parameter vector,
It is important to note that if σ is restricted to (0,1), then (6) can be considered as a special case of (4).
The pdf corresponding to (6) is
(7)
3 Sub-models description
This section provides some sub-models from NEHT-F family called the NEHTW, NEHT-normal (NEHTN), and NEHTW-gamma (NEHTG) distributions. These sub-models can provide bimodal, unimodal, symmetric and asymmetric density shapes, and constant, increasing, modified bathtub, decreasing, J-shaped, and reversed-J shaped hazard rate functions.
3.1 NEHTW distribution
Consider the rv X that follows the one-parameter Weibull distribution with cdf , and pdf
. Then, the cdf of the NEHTW distribution takes the form
(8)
The corresponding pdf and hrf of the NEHTW model are
(9)
and
(10)
The QF of the NEHTW model is given by
(11)
where 0 < p < 1.
The one parameter Weibull distribution with parameter α follows as a special sub-model from the NEHTW distribution when η = σ = 1. For σ = 1, the two-parameter Weibull distribution follows as a special sub-model from the NEHTW model. The exponential distribution follows as a special case for α = σ = 1. For σ = 1 and α = 2, the NEHTW model reduces to the Rayleigh distribution.
For selected parameters values of the NEHTW distribution, some possible shapes for its density and hazard functions are sketched in Fig 1.
3.2 NEHTN distribution
Consider the rv X that follows the normal distribution with cdf where
, and pdf
where
and
. Then, the cdf of the NEHTN distribution takes the form
(12)
The corresponding pdf and hrf of the NEHTN model are
(13)
and
(14)
For selected parameters values of the NEHTN distribution, some possible shapes for its density and hazard functions are sketched in Fig 2, the shape of the NEHTN distribution can be unimodal, bimodal, right skewed or symmetry.
3.3 NEHTG distribution
Consider the rv X that follows the gamma (G) distribution with cdf where
is the G function and
is the lower incomplete G function, and pdf
where x > 0, α, β > 0. Then, the cdf of the NEHTG distribution takes the form
(15)
The corresponding pdf and hrf of the NEHTG model are
(16)
and
(17)
For selected parameters values of the NEHTG distribution, some possible shapes for its density and hazard functions are sketched in Fig 3, the shape of the NEHTG distribution can be unimodal, bimodal, right skewed, symmetry or decreasing.
4 Two risk measures and numerical results
The actuaries are often interested in evaluating the exposure to market risks appear due to changes in some underlying variables including prices of equity, exchange rates or interest rates. This section is devoted to determining two important risk measures namely, VaR and TVaR of the proposed NEHTW distribution, which are useful in portfolio optimization.
The VaR is known in actuarial sciences as the quantile premium principle or quantile risk measure and it has been adopted by practitioners as a standard financial market risk. The VaR of a rv X is defined by the qth quantile of its cdf and it may be specified with a certain degree of confidence, q which is typically can be 90%, 95% or 99%, as shown in [28]. If X has cdf (8), then the VaR of X is given by
(18)
where q is the qth quantile of X.
The TVaR is an important measure to determine the expected value losses given that an event outside a certain probability level is occurred. Hence, the TVaR of the NEHTW model is specified by
(19)
Inserting (9) in Eq (11), we can write
In the next section, based on (10) and (19), we provide a simulation study of VaR and TVaR to show empirically the heaviness of the tail of the NEHTW distribution.
In this section, numerical simulation results were presented for the VaR and TVaR of the two-parameter Weibull and the NEHTW distributions for several values of their parameters. The process was described below as follows:
- Generating random samples of size n = 100 from both Weibull and NEHTW distributions.
- Estimating the parameters of both models by the maximum likelihood approach.
- The VaR and TVaR of both distributions were calculated using 1000 repetitions.
The numerical values of the VaR and TVaR were reported in Tables 1 and 2. For visual comparison of the results in Tables 1 and 2, the graphs for the VaR and TVaR of the Weibull and NEHTW distributions are sketched in Fig 4.
The model with greater values of both VaR and TVaR is said to have a heavier tail. The simulation values in Tables 1 and 2 illustrate that the NEHTW distribution, with higher values of both VaR and TVaR, has a heavier tail than Weibull distribution. This important result can be shown graphically in Fig 4.
5 Properties of the NEHT-G family
Some mathematical quantities of the NEHT-G family were introduced in this section.
5.1 Linear representation
Using the two power series
(21)
and
(22)
The NEHT-G density can be expressed as
Hence, the pdf of the NEHT-G family takes the form
(23)
where h(j+k+1)(x) is the exponentiated-G (exp-G) pdf with positive power parameter (j + k + 1) and
Hence, the density function of the NEHT-G family can be expressed as a linear combination of exp-G densities. So, Eq (23) reveals that some mathematical quantities of the proposed family can be derived from the quantities of the exp-G family. Hereafter, let Yj+k+1 denote a rv having the exp-G density with power parameter (j + k + 1), then some properties of the rv X are expressed simply from those of Yj+k+1.
5.2 Moments and generating function
The nth moment of the NEHT-G family was expressed from (23) as
The rth incomplete moment of the NEHT-G family follows simply from (23) as
(24)
The first incomplete moment (FIM) of X follows from (23) as
where
is the FIM of the exp-G family which can be determined numerically or analytically from a baseline qf QG(u) = G−1(u;ξ).
The moment generating function (mgf) of the NEHT-G family was derived from (23) as
where Mj+k+1(t) is the mgf of Yj+k+1 and
. Then, M(t) follows from the exp-G mgf.
5.3 Rényi entropy
The Rényi entropy (RE) of a rv X is a measure of variation of uncertainty. The RE is given by
Using the NEHT-G density, we obtain
By applying the two power series in (21) and (22), we have
5.4 Order statistics
Let X1, …, Xn be a random sample from the NEHT-G family and consider their associated order statistics, X1:n, …, Xn:n. Then, the pdf of the sth order statistic Xs:n, gs:n(x), has the form
Using the cdf and pdf of the NEHT-G class, the pdf of the Xs:n reduces to
where φ = η(n + r − s + 1). Applying the two power series in (21) and (22) to the above equation, we obtain
Hence, the pdf of the sth order statistic for the NEHT-G family is a linear combination of exp-G densities as follows
(25)
where hj+k+1(x) is the pdf of the exp-G class with power parameter (j + k + 1) and
Using (25), some mathematical quantities of Xs:n follows simply from the properties of Yj+k+1.
6 Estimations
To estimate the parameters of the NEHT-F family, this section assigns the maximum likelihood estimators (MLEs) to estimating and to provide some simulations to explore the behavior of the MLEs.
6.1 Maximum likelihood estimation
Suppose that x1, x2, …, xn are given values of a random sample of size n from the NEHT-F family with parameters (η, σ, ξ). The log–likelihood function takes the form
(26)
where Θ = (η, σ, ξ⊤)⊤.
The partial derivatives of (26) are
(27)
(28)
and
(29)
By setting ,
, and
one can solve them numerically to obtain the MLEs of η, σ and ξ, respectively.
These nonlinear system of equations can be solved using any statistical software.
7 Eight estimation methods for the NEHTW parameters
Eight estimation methods have been opted in this section to estimate the NEHTW parameters, namely: weighted least-squares (WLSE), ordinary least-squares (OLSE), maximum likelihood (MLE), maximum product of spacing (MPSE), Cramér-von Mises (CVME), Anderson-Darling (ADE), right-tail Anderson-Darling (RADE), and percentile estimator (PCE).
Suppose that x1, x2, …, xn are given values of a random sample of size n from the NEHTW distribution with parameters α, σ and η.
7.1 Maximum likelihood
The log-likelihood function for the NEHTW model in (9) is given by
(30)
where Θ = (α, σ, η)⊤. The function provided in (30) can be numerically solved by using Newton-Raphson method (iteration method). The partial derivatives of (30), with respect to the parameters, are
and
7.2 Ordinary and weighted least-squares
The OLSE of the NEHTW parameters can be obtained by minimizing the following function with respect to α, σ and η,
Further, the OLSE of the NEHTW parameters can be obtained by solving the non-linear equation
where
and
(31)
The WLSE of the NEHTW parameters are obtained by minimizing the following
with respect to α, σ and η. Also, the WLSE can be obtained by solving the non-linear equation
where Δ1(⋅|α, σ, η), Δ2(⋅|α, σ, η) and Δ3(⋅|α, σ, η) are defined in Eq (31).
7.3 Maximum product of spacing
The MPSE is considered a good alternative to the maximum likelihood method. Let Di(α, σ, η) = F(x(i)|α, σ, η) − F(x(i−1)|α, σ, η), for i = 1, 2, …, n + 1, be the uniform spacing of a random sample from the NEHTW model, where F(x(0)|α, σ, η) = 0, F(x(n+1)|α, σ, η) = 1 and . The MPSE of the NEHTW parameters can be obtained by maximizing the “geometric mean of the spacing”
with respect to α, σ and η, or by maximizing the “logarithm of the geometric mean” of sample spacings
Also, the MPSE can be obtained by solving the following nonlinear equations
where Δ1(⋅|α, σ, η), Δ2(⋅|α, σ, η) and Δ3(⋅|α, σ, η) are defined in Eq (31).
7.4 Cramér-von mises
The CVME of the NEHTW parameters are obtained by minimizing
with respect to α, σ and η. Also, the CVME can be numerically obtained by solving the following non-linear equation
where Δ1(⋅|α, σ, η), Δ2(⋅|α, σ, η) and Δ3(⋅|α, σ, η) are defined in Eq (31).
7.5 Anderson-Darling and right-tail Anderson-Darling
Suppose that x(1), x(2), …, x(n) is the ordered random sample from F(x) of the NEHTW model. The ADE of the NEHTW parameters can be obtained by minimizing
or by solving the non-linear equation
Also, the RADEs of the NEHTW parameters can be obtained by minimizing
with respect to β, θ and a, which equivalently by solving the non-linear equations
where Δ1(⋅|α, σ, η), Δ2(⋅|α, σ, η) and Δ3(⋅|α, σ, η) are defined in Eq (31).
7.6 Percentile
From (11), the PCE of the parameters of NEHTW model can be obtained by minimizing the following function
with respect to α, σ and η.
8 Simulation study
To assessing the performance of the eight estimates aforementioned for the NEHTW parameters, we devote this section.
The simulation results is carried out for the NEHTW distribution by the function optim() with the argument method = “L-BFGS-B” available by R software (version 4.0.5) [29]. We generated N = 4000 samples of five sizes, where n = {50, 80, 120, 200, 300} from the NEHTW distribution for different values of the parameters η = {0.40, 1.60, 2.75}, σ = {0.50, 1.75, 3.00} and α = {0.75, 1.50, 4.00}. The MSEs and biases can be computed, for Θ = (η, σ, α)⊤, by the formulas: average of absolute value of biases (),
, average of mean square errors (MSEs),
, and average of mean relative errors (MREs),
.
Using the eight estimators aforementioned with each parameter combination and each sample, the NEHTW parameters are estimated, after that the Bias, MSEs and MREs of the parameter estimates are computed. To save more space, five out of twenty seven simulated outcomes are listed in Tables 3–7, one can easily observe that when the sample size increases, for all parameter combinations except the PCE method. The PCE shows the property of consistency for all parameter combinations, except the combinations Θ = (η = 0.40, σ = 1.75, α = 1.50)⊤ for the parameter σ, and Θ = (η = 0.40, σ = 0.50, α = 0.75)⊤, Θ = (η = 0.40, σ = 1.75, α = 0.75)⊤ and Θ = (η = 0.40, σ = 3.00, α = 0.75)⊤, for the parameter α.
In addition, Tables 3–7 show the rank of each estimator among all the remaining estimators in each row, with partial sum of the ranks for each column (∑ Ranks) for each sample size.
The partial and overall ranks of all estimators are shown in Table 8. From Table 8, we can conclude that the MPSE outperforms all the other estimators with an overall score of 142. Thus, for the NEHTW distribution we recommend the MPSE estimation method.
9 Applications and numerical computations of VaR and TVaR
The key application of the heavy-tailed distributions is the insurance loss phenomena or extreme value theory. Here, we consider two data sets from insurance sciences. The flexibility of the NEHTW model was illustrated by analyzing two insurance data sets. Furthermore, corresponding to the analysed data sets, we calculate the VaR and TVaR of the NEHTW and Weibull models to show empirically, using the two real data sets, that NEHTW model has a heavier tail than, its sub-model, Weibull distribution.
The comparison of the NEHTW distribution is made with some important distributions including two-parameter Weibull, Lomax, Burr-XII (BX-II), exponentiated Lomax (EL) and beta Weibull (BW) distributions.
To decide about the best fitting among the competitive distributions, certain comparative tools called, AIC, BIC, Cramer Von Mises (CM), Anderson Darling (AD), and KS statistic with its corresponding p-value are taken in to account. The formulae of these discrimination measures can be explored in [30].
9.1 Applications to earthquake insurance and vehicle insurance losses data
In this subsection, we consider two practical insurance applications to show the flexibility of the NEHTW model in practice.
9.1.1 The earthquake insurance data.
We begin with our first illustration by considering a heavy-tailed data set which refers to the earthquake insurances. This data set is available online at: https://earthquake.usgs.gov/earthquakes. The MLEs with standard errors (in parentheses) of the parameters and the estimated values of the considered measures were reported in Tables 9 and 10. Based on the values in Table 10, we conclude that the NEHTW model provide adequate fits as compared to other competitors. So, we prove empirically that NEHTW distribution can be a better model than other competitive models. Furthermore, corresponding to the earth quick insurance data, the estimated pdf and cdf of the NEHTW model were displayed in Fig 5. The Kaplan Meier survival (KMS) and PP plots are presented in Fig 6. The plots show that the NEHTW model has the best fitting to the earthquake insurance data.
9.1.2 The vehicle insurance loss data.
The second data set is about vehicle insurance losses and it is available online at the following URL: https://data.world/datasets/insurance. The parameters estimates and associated standard error (in parentheses) of the NEHTW model and other models were listed in Table 11. The values of analytical measures for the NEHTW and other studied models were reported in Table 12. The values in Table 12 show that the NEHTW model outperforms all fitted models. The fitted pdf and cdf of the NEHTW model were shown in Fig 7. The PP plot and KMS plot of the NEHTW model were displayed in Fig 8. It is noted, from Fig 8, that the NEHTW model has a close fits to the PP and KMS plots.
9.2 The VaR and TVaR based on the two insurance data sets
This subsection is devoted to exploring the VaR and TVaR measures of the NEHTW and Weibull distributions via the estimated parameters of the two analyzed insurance data sets previously discussed in Section 7.1. The empirical results were listed in Tables 13 and 14 and they are displayed graphically in Fig 9.
The results for the two actuarial measures, Var and TVaR, of the NEHTW and Weibull distributions, which were reported in Tables 13 and 14, show that the NEHTW distribution has a heavier tail than the tail of Weibull distribution and hance it can be used for modelling heavy-tailed insurance data.
10 Conclusions
In this article, a new family of heavy-tailed distributions is proposed and studied. A special sub-model of the introduced family called, the new extended heavy-tailed Weibull (NEHTW) distribution is discussed in detail. The introduced NEHTW model is very flexible and can be adopted effectively to model data with heavy tails which encountered in insurance science. The maximum likelihood and other seven estimation approaches are adopted to estimate the NEHTW parameters. Their performances are assessed using detailed simulation results which are ranked to explore the best estimation approach for the model parameters. Based on our study, the maximum product of spacing is recommended to estimate the NEHTW parameters. Two important risk measures of the NEHTW model are studied numerically and empirically using a real-life insurance data. Both numerical and empirical studies of the two risk measures show that the introduced NEHTW distribution has heavier tail than its sub-model and can be used quite effectively in modelling heavy tailed insurance data sets. Finally, practical applications to insurance data sets are analyzed and the comparisons of the NEHTW model are made with some other important competitors. The practical applications and simulation of the two actuarial measures show that the NEHTW distribution is a good alternative for modelling heavy-tailed insurance data.
There are some possible future extensions to this study, such as a discrete version of the proposed family can be introduced following the work of [31]. Furthermore, a survival regression extended heavy-tailed Weibull model can be addressed for complete and censored data.
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